MinT - Best Known (75, 75+75, s)-Nets in Base 5

Best Known (75, 75+75, s)-Nets in Base 5

(75, 75+75, 94)-Net over F₅ — Constructive and digital

Digital (75, 150, 94)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 44, 22)-net over F₅, using
  - net from sequence [i] based on digital (7, 21)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 106, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(75, 75+75, 145)-Net over F₅ — Digital

Digital (75, 150, 145)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+75, 2364)-Net in Base 5 — Upper bound on s

There is no (75, 150, 2365)-net in base 5, because

1 times m-reduction [i] would yield (75, 149, 2365)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 142 097159 399288 263808 763628 389542 356718 062112 750702 593630 688475 697288 670711 101548 421312 599204 906827 049381 > 5¹⁴⁹ [i]